Abstract and Applied Analysis
Volume 2003 (2003), Issue 8, Pages 479-502
doi:10.1155/S1085337503212033

Attractors of iterated function systems and Markov operators

Józef Myjak1,2 and Tomasz Szarek1,3

1Dipartimento di Matematica Pura ed Applicata, Università di L'Aquila, Via Vetoio, L'Aquila 67100, Italy
2AGH University of Science and Technology, Mickiewicza Avenue 30, Kraków 30-059, Poland
3Institut of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice and Department of Mathematics, Technical University of Rzeszów,W. Pola 6, Rzeszów 35-959, Poland

Received 22 December 2001

Copyright © 2003 Józef Myjak and Tomasz Szarek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper contains a review of results concerning “generalized” attractors for a large class of iterated function systems {wi:i∈I} acting on a complete separable metric space. This generalization, which originates in the Banach contraction principle, allows us to consider a new class of sets, which we call semi-attractors (or semifractals). These sets have many interesting properties. Moreover, we give some fixed-point results for Markov operators acting on the space of Borel measures, and we show some relations between semi-attractors and supports of invariant measures for such Markov operators. Finally, we also show some relations between multifunctions and transition functions appearing in the theory of Markov operators.